Expand the Logarithmic Expression natural log of ((3x)/(y^2))^4

$\ln ({(\frac{3 x}{y^{2}})}^{4})$

Step 1

Expand $\ln ({(\frac{3 x}{y^{2}})}^{4})$ by moving $4$ outside the logarithm.

$4 \ln (\frac{3 x}{y^{2}})$

Step 2

Rewrite $\ln (\frac{3 x}{y^{2}})$ as $\ln (3 x) - \ln (y^{2})$ .

$4 (\ln (3 x) - \ln (y^{2}))$

Step 3

Expand $\ln (y^{2})$ by moving $2$ outside the logarithm.

$4 (\ln (3 x) - (2 \ln (y)))$

Step 4

Multiply $2$ by $- 1$ .

$4 (\ln (3 x) - 2 \ln (y))$

Step 5

Rewrite $\ln (3 x)$ as $\ln (3) + \ln (x)$ .

$4 (\ln (3) + \ln (x) - 2 \ln (y))$

Step 6

Apply the distributive property.

$4 \ln (3) + 4 \ln (x) + 4 (- 2 \ln (y))$

Step 7

Multiply $- 2$ by $4$ .

$4 \ln (3) + 4 \ln (x) - 8 \ln (y)$